330 HlSTOKY OF THE THEOHT OF NuMBEES. [CHAP. XI If /(x)=log x, the second function (1) becomes v(x), having the value* log p when a: is a power of the prime p, otherwise the value 0. Besides the resulting formulas, others are found by taking f(x)=v(x), Jacobi's symbol (A/z) in the theory of quadratic residues, and finally the number of representations of x by the system of quadratic forms of discriminant A. L. Saint-Loup13 represented graphically the divisors of a number. Write the first 300 odd numbers in a horizontal line; the 300 following numbers are represented by points above the first, etc. Take any prime as 17 and mark all its multiples; we get a rectilinear distribution of these multiples, which are at the points of intersection of two sets of parallel lines. J. Hacks14 proved that the number of integers ^m which are divisible by an nth power >1 is where the k's range over the primes >1 [Bougaief6]. Then ^2(w) = m—p2(m) is the number of integers ^m not divisible by a square >1, and A like formula holds for \l/3 = m — p^(m), using quotients of m by cubes. L. Gegenbauer140 found the mean of the sum of the reciprocals of the fcth powers of those divisors of a term of an unlimited arithmetical progression which are rth powers ; also the probability that a term be divisible by no rth power; and many such results. L. Gegenbauer16 noted that the number of integers 1, . . . , n not divisible by a Xth power is (2) Qx(n)=2 MX). z-iL-C J Ch. de la Valle"e Poussin16 proved that, if x is divided by each positive number ky+b^x, the mean of the fractional parts of the quotients has for x= oo the limit 1 — C; if x is divided by the primes ^x, the mean of the fractional parts of the quotients has f or x = oo tiie limit 1 — C. Here C is Euler's constant.8 L. Gegenbauerllr proved, concerning Diriehlet's3 quotients Q of the remainders (found on dividing n by 1, 2, . . . , n in turn) by the corresponding divisors, that the number of Q's between 0 and 1/3 exceeds the number of Q's between 2/3 and 1 by approximately 0.1 79n, and similar theorems. *Cf. Bougaief" of Ch. XIX. "Comptes Rendua Paris, 107, 1888, 24; ficole Norm. Sup., 7, 1890, 89. "Acta Math., 14, 1890-1, 329-336. "'Sitzungsber. Ak. Wien (Math.), 100, Ila, 1891, 1018-1053. »/Wd., 100, 1891, Ila, 1054. Denkschr. Akad. Wien (Math.), 49 I, II, 1885; 50 1, 1885. Cf. Gegenbauer79 of Ch. X. "Annales de la soc. sc. Bruxelles, 22, 1898, 84-90. 1T8itzungsberichte Ak. Wisa. Wien (Math.), 110, 1901, Ila, 148-161.